# 30-60-90 and 45-45-90 Triangle Exploration

Investigation 1
1. Given the equilateral triangle Above, draw an altitude from point C to side AB. This new point will be called D.

2. What happens to side AB when the altitude is drawn in?

3. What are the measures of angles CAB and CBA? (Hint: there is an angle measure tool-fourth from the left)

4. What are the measures of angle ACD and BCD?

5. In the two right triangles, which sides are the shortest? Explain how you determined this.

6. In the two right triangles, which sides are the longest? Explain how you determined this.

7. How do the smallest sides of the right triangles compare to the longest sides of the right triangles?

8. Label the smallest sides of the right triangles with the same variable. (The second to the right button has a text tool).

9. Using the variable from #8, write an expression for the longest side of each right triangle.

10. Find an expression for the attitude (CD)

11. In mathematics, 30-60-90 triangles are one of the special right triangles that are often used. How are the sides of a 30-60-90 triangle related to each other? Why would this relationship be true for all 30-60-90 Triangles?

Investigation 2
1. Draw the Diagonal AC

2. What type of triangles are ACD and ACB? Explain how you determined this answer.

3. What are the measures of angles DAC and ACD? Explain how you determined this answer. Label these measurements on the picture.

4. What are the measures of angles BAC and BCA? Explain how you determined this answer. Label these measurements on the picture.

5. Label the legs of Triangle ACD with the same variable.

6. Using the variable from #5, write an expression for the hypotenuse of Triangle ACD.

7. In mathematics, 45-45-90 triangles are one of the special right triangles that are often used. How are the sides of a 45-45-90 triangle related to each other? Why would this relationship be true for all 45-45-90 triangles?